Exercises: Reliability, Testing, and Why Rockets Fail

Work these with a calculator; use $g_0 = 9.81\ \text{m/s}^2$ where propulsion appears, and treat all reliabilities as independent unless a problem says otherwise. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. For "implement it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does.

Part A — Warm-ups: compute a reliability (⭐)

32.1 † Three components in series have reliabilities $0.99$, $0.95$, and $0.98$. What is the system reliability, and what is the probability of mission loss?

32.2 An avionics box has an MTBF of $500{,}000\ \text{hours}$. Using $R(t) = e^{-\lambda t}$, find its reliability over a 2-year mission ($1\ \text{yr} = 8{,}766\ \text{hr}$).

32.3 † Two identical redundant units, each $R = 0.90$, operate in active parallel. What is the reliability of the pair?

32.4 A single-string subsystem has five components in series, each $R = 0.98$. What is the subsystem reliability? By how much did the series multiplication pull it below the weakest link?

32.5 † A mission is quoted a "probability of loss" of $0.08$. What is its reliability? State the function/conditions/duration that this number is meaningless without.

32.6 Three identical units, each $R = 0.80$, are placed in parallel. What is the system reliability? How many "nines" of unreliability did the third unit buy over a single unit?

32.7 † A critical chain has four functions in series: $0.97$, $0.92$, $0.99$, $0.95$. (a) Compute the series reliability. (b) Identify the weakest link. (c) Add one redundant copy of the weakest link and recompute the system reliability. (d) How much did the mission-loss probability drop?

32.8 For $n$ identical parallel units, show algebraically that $R_{\text{parallel}} = 1 - (1-R)^n \to 1$ as $n \to \infty$ for any $0 < R < 1$, and explain in one sentence what physical fact makes this not a license to ignore reliability by piling on backups.

32.9 † A function must reach $R \ge 0.95$, but the best single string available is only $R = 0.85$. How many identical strings in parallel are required? Show the inequality you solve.

Part C — Implement it in Python (⭐⭐)

Write each function, hand-trace it for the given inputs, and record the result in an # Expected output: comment. Do not run it.

32.10 † Write series(rs) returning the product of a list of reliabilities. Trace it for series([0.99, 0.98, 0.97, 0.96]).

32.11 Write redundant(r, n) returning $1-(1-r)^n$. Trace it for redundant(0.85, 3).

32.12 † Write k_of_n(r, n, k) returning the probability that at least $k$ of $n$ identical units survive (use math.comb). Trace it for k_of_n(0.98, 4, 3) — a system of four units that tolerates one failure.

Part D — Find the error (⭐⭐)

32.13 † A student sizes a redundant pair of flight computers, each $R = 0.99$, as $0.99 \times 0.99 = 0.9801$, and concludes that "adding a backup made it worse." What did they actually compute, what is the correct formula for redundancy, and what is the right answer?

32.14 A design claims two redundant reaction wheels give $1-(1-0.90)^2 = 0.99$ reliability — but both wheels draw from a single shared power converter of reliability $0.97$. Explain the common-cause flaw, and estimate the real reliability of the "redundant" wheel function.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

32.15 † Reliability allocation. A mission targets system reliability $R = 0.90$ across four subsystems in series. (a) If the four are equally reliable, what $R$ must each meet? (b) If one subsystem can only achieve $R = 0.95$, what average reliability must the other three then have to still hit $0.90$?

32.16 (Mission project) Build the FMEA-lite for your chosen mission (Track A/B/C/D): list your five highest-risk items with failure mode, effect, detection, a severity (1–4) × likelihood (1–5) score, and a mitigation for each HIGH-risk row. Add it to your MDR risk assessment.

32.17 † Reliability per kilogram. A spacecraft has series reliability $0.82$ and $60\ \text{kg}$ of mass margin. Three candidate redundancies compete: a dual computer (block $0.90 \to 0.99$, $+8\ \text{kg}$), a redundant reaction-wheel set (block $0.93 \to 0.995$, $+12\ \text{kg}$), and a redundant valve (block $0.96 \to 0.998$, $+3\ \text{kg}$). Rank them by reliability gained per kilogram (use the relative system gain, new-block/old-block, per kg). Can you afford all three?

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

32.18 † A launch vehicle has roughly 200 flight-critical "must-work" events (ignitions, separations, deployments). If the program wants a 95% launch reliability and treats the events as an independent series, what average per-event reliability does that demand? Comment on what the number implies about workmanship.

32.19 Why can't you just make every subsystem redundant and be done with reliability? Frame your answer in terms of mass and the rocket equation of Chapter 3.

32.20 † Why can't you just test one spacecraft exhaustively until it never fails, and then fly that exact unit? Use the qualification-vs-acceptance distinction in your answer.

Part G — Interleaved & synthesis (from Chapters 29, 7, 3, 31)

32.21 † (Ch. 29) In Chapter 29 you learned that mass margin is "bought down" from 25–30% at MDR to near zero at launch. Relate this buy-down to reducing uncertainty, and explain why holding margin constant would be the wrong policy.

32.22 (Ch. 7) A re-entry heat shield is a catastrophic single point of failure. Explain why hardware redundancy (a second shield) is impractical, and name the strategy that replaces redundancy for such an item.

32.23 † (Ch. 3, synthesis) Adding a redundant engine to an upper stage costs $400\ \text{kg}$. Using the rocket-equation intuition (upper-stage mass is "payload" to every stage below), explain why buying reliability with redundancy is especially expensive high in the stack.

32.24 (Ch. 31, synthesis) During the Mars Climate Orbiter cruise, controllers noticed the spacecraft tracking off-course but did not resolve it before arrival. Relate this to the "detection" column of an FMEA: why is detection without a response effectively worthless?

32.25 (Synthesis) Two rockets each fly 20 times with no failures. One was designed to $R = 0.90$, the other to $R = 0.99$. Can you distinguish them from this record? Compute the probability of 20 clean flights for each, and comment on what it takes to demonstrate high reliability by flight history alone.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: correct series/parallel set-up, explicit identification of single points of failure and common-cause risks, units on every physical quantity, and a sanity check on the final number.